Question

Let $x,y$ and $z$ be the respective sum of the first $n$ terms, the next $n$ terms and the next $n$ terms of a geometric progression, then $x,y,z$ are in

• A. arithmetic progression
• B. geometric progression
• C. harmonic progression
• D. None of these

Answer 1

Answer: (B)

Let $A$ be the 1^st term and $R$ be the common ratio of the given G.P.
Sum of the first $n$ terms of the G.P. is $x=\frac{A(1-R^n)}{1-R}$ . The next $n$ terms form a G.P. series of $n$ terms with the first term as the ${\left(n+1\right)}^{th}$ term of the given G.P. is given by $t_{n+1}=A{R}^n$ . Sum of the next $n$ terms of the G.P. is $y\Rightarrow y=\frac{A{R}^n(1-R^n)}{1-R}$ and
$z=A{R}^{2n}\frac{\left(1-R^n\right)}{1-R}$ $\Rightarrow y^2=zx$
$\therefore$ $x$ , $y$ , $z$ are in G.P.